In enriched homotopy theory, the presentation is given by an enriched model category or an enriched homotopical category, and it may presents an “enriched (∞,1)(\infty,1)-category” or be a more powerful presentation of a bare (∞,1)(\infty,1)-category (notably if the enrichment is over sSet). In the enriched category theory context the appropriate notion is a weighted homotopy limit, which may construct “weighted (∞,1)(\infty,1)-limits” in the presented “enriched (∞,1)(\infty,1)-category” or may be a more powerful tool for constructing plain (∞,1)(\infty,1)-categorical limits (in particular if the enrichment is over sSet). Note that as yet, no fully general notion of “enriched (∞,1)(\infty,1)-category” exists; see homotopical enrichment.

The ordinary limit and colimit operations on DD-diagrams are (as described there) the right and left adjoints of the functor const:C→[D,C]const : C \to [D,C], or equivalently left and right Kan extension along the unique functor !:D→*!\colon D\to * to the terminal category.

The (globally defined) homotopy limit and colimit are accordingly the right and left homotopy Kan extension along !:D→*!\colon D\to *:

The homotopy limit of a functor F:D→CF : D \to C is, if it exists, the image of FF under the right derived functor of the limit functor limD:[D,C]→Clim_D : [D,C] \to C with respect to the given weak equivalences on CC and the objectwise weak equivalences on [D,C][D,C]:

holimDF:=(ℝlimD)F.
holim_D F := (\mathbb{R} lim_D)F
\,.

The homotopy colimit of a functor F:D→CF : D \to C is, if it exists, the image of FF under the left derived functor of the colimit functor colimD:[D,C]→Ccolim_D : [D,C] \to C with respect to the given weak equivalences on CC and the objectwise weak equivalences on [D,C][D,C]:

hocolimDF:=(𝕃colimD)F.
hocolim_D F := (\mathbb{L} colim_D)F
\,.

Alternative definitions can be formulated at the level of the homotopy category W−1CW^{-1} C one defines a localized version Δ¯I:W−1C→WI−1CI\bar{\Delta}^I : W^{-1} C\to W_I^{-1} C^I of the diagonal functor ΔI:C→CI\Delta^I : C\to C^I and define the homotopy limits and colimits as the adjoints of Δ¯I\bar{\Delta}^I (at least at the points where the adjoints are defined). Here WI⊂Mor(CI)W_I\subset Mor(C^I) are the morphisms of diagrams whose all components are in W⊂Mor(C)W\subset Mor(C). The above definitions via derived functors (Kan extensions) follow once one applies the general theorem that the derived functors of a pair of adjoint functors are also adjoint and noticing that (ΔI,Δ¯I)(\Delta^I,\bar{\Delta}^I) is a morphism of localizers (and in particular that Δ¯I\bar{\Delta}^I with the identity 2-cell is a Kan extension (simultaneously left and right)).

In the enriched case, this must be suitably modified to deal with weighted limits as well as enrichment of both CC and DD.

by precomposition with a cofibrant replacement functor (for the colimit) and a fibrant replacement functor (for the limit).

Local definition

The local definition requires making precise the notion of a homotopy commutative cone on a diagram.

For the case of SimpSet-enrichment one elegant way to do so is in terms of suitable weighted limits as described in the example section at weighted limit: a homotopy commutative cone with tip c∈Cc \in C on a diagram F:K→CF : K \to C in an SimpSet\Simp\Set-enriched category CC is a natural transformation W⇒C(c,F(−)):K→SimpSetW \Rightarrow C(c,F(-)) : K \to \Simp\Set where the weight functor WW is not constant on the point, as for ordinary limits, but is given by W:k↦N(K/k)W : k \mapsto N(K/k).

The same idea works if we are enriched over a category VV that is not SimpSet\SimpSet but is itself enriched over SimpSet\Simp\Set, such as topological spaces or spectra, since then any VV-category becomes a SimpSet\Simp\Set-category as well in a natural way. Finally, although a general model category need not be enriched over anything, it is always “almost” enriched over SimpSet\Simp\Set, and so one can still make sense of this using the techniques of framings and resolutions; see the books of Hirschhorn and Hovey.

Following the reasoning described in Example 1 of representable functor one then defines the homotopy limit LL of a functor F:K→CF: K \to C to be a representing object for such homotopy cones, in the sense that we have a (weak) equivalence

Map(X,L)≃HoCones(X,F) Map(X,L) \simeq HoCones(X,F)

of hom-objects (spaces or simplicial sets in the classical context; enriched hom-objects in the enriched context).

Global versus local

The global definition is formulated in terms of weak equivalences only, while the local definition is formulated in terms of homotopies only. However, in practical cases, derived functors exist because their input objects (in this case, the diagram FF) can be replaced by “good” (fibrant and/or cofibrant) objects in such a way that weak equivalences become homotopy equivalences. The derived functor of limlim at the input object FF is then computed by applying the ordinary functor limlim to a good replacement RFR F of FF.

It then turns out that the “good” (precisely, “fibrant”) replacement RFR F “builds in” precisely the right homotopies so that an ordinary cone on RFR F is the same as a homotopy-commutative cone on FF. Therefore, lim(RF)lim (R F), which is the global homotopy-limit of FF, is a representing object for homotopy-commutative cones on FF, and thus is also a local homotopy-limit of FF. There is a dual argument for colimits using cofibrant replacements.

Formal versions of this argument can be found in many places. Perhaps the original statement can be found in XI.8.1 of:

(As was often the case with Kan’s papers at that time, there are some details omitted from that treatment, but most are, as he claimed, quite easy to complete.) For another approach in an algebraic context, there is a description in Illusie’s thesis.

Strictness

Another notable difference between the local and global definitions is that the global definition can only ever define the homotopy limit up to weak equivalence (isomorphism in the homotopy category), while in the local definition we could require, if we wanted to, an actual isomorphism

Map(X,L)≅HoCones(X,F) Map(X,L) \cong HoCones(X,F)

of hom-objects, rather than merely a weak equivalence. By analogy with strict 2-limits, we may call such an object a strict homotopy limit.

Frequently a strict homotopy limit does in fact exist, and can be constructed as a weighted limit in the ordinary (enriched) category in question. In such cases, the strict homotopy limit may be easier to compute with than an arbitrary homotopy limit merely known to have the up-to-weak-equivalence universal property. Thus, sometimes when people say the homotopy limit they refer mean a strict homotopy limit.

When a strict homotopy limit exists, an arbitrary homotopy limit may be defined as an object which is (weakly) equivalent to the strict homotopy limit.

Derivators

It is noteworthy that the homotopy limit and colimit in a category with weak equivalences are drastically different from the ordinary limit and colimit in the corresponding homotopy category: the universal property of the full (∞,1)(\infty,1)-categorical limits and colimits crucially does take into account the explicit higher cells and does not work just up to any higher cells.

This (obvious) observation is a very old one and can be seen to be one of the driving forces behind the insight that just working with homotopy categories misses crucial information, something which today is understood as the statement that a homotopy category is just the decategorification of an (∞,1)-category.

While the full theory of (∞,1)-categories is one way to impose the correct notion of higher categorical limits, there are other ways to deal with issue. Due to Alexander Grothendieck is the technique of using derivators for keeping track of homotopy limits.

Roughly, the idea of a derivator is that while the single homotopy categoryHo(C)Ho(C) of a category CC with weak equivalences is not sufficient information, the homotopy limit of a diagram in [D,C][D,C]is encoded in the homotopy category Ho([D,C])Ho([D,C]) of that functor category (this is after all the domain of the plain 1-categorical derived functor of the limit functor). Accordingly, what is called the derivator of a category with weak equivalences CC is a collection of all the homotopy categories Ho([D,C])Ho([D,C]) of all the possible diagram categories [D,C][D,C], as DD runs over all small categories. See there for more details.

Computational methods

Above we defined homotopy (co)limits in general. There are various more specific formulas and algorithms for computing homotopy (co)limits. Here we discuss some of these

Qproj:[D,C]→[D,C]Q_{proj} : [D,C] \to [D,C] be a cofibrant replacement for the projective model structure on functors, so that for any diagram FF the diagram QprojFQ_{proj} F is a projectively cofibrant diagram (see there for more details). Then the homotopy colimit is presented by the ordinary colimit on QprojFQ_{proj} F:

This is sometimes called the Quillen formula for computing homotopy colimits. Analogously with PinjP_{inj} a fibrant replacement functor for the injective model structure, we have

holimF≃limPinj(F).
holim F \simeq lim P_{inj}(F)
\,.

Often, however, it is inconvenient to produce a resolution of a diagram. Because often all the work is in finding the resolution, while it is easy to evaluate the original functor on it. Therefore one wants ways to slightly change the setup of the problem such that the computation of the resolutions becomes more systematic. One such way is the use of derived (co)ends, discussed below.

Resolved (co)ends

Let Q′projQ'_{proj} a cofibrant replacement functor for [Dop,sSetQuillen]proj[D^{op}, sSet_{Quillen}]_{proj} (notice the opposite category) and QinjQ_{inj} one for [D,C]inj[D,C]_{inj}. Let *∈[Dop,sSet]* \in [D^op,sSet]simplicial presheaf constant on the terminal simplicial set and Q′proj(′)Q'_{proj}(') its projective cofibrant replacement.

If DD happens to be a Reedy model category we can equivalently use in this expression also the Reedy model structures [D,C]Reedy[D,C]_{Reedy} and [Dop,C]Reedy[D^{op}, C]_{Reedy} and obtain the homotopy colimit as

This formula is sometimes called the Bousfield-Kan formula (see also Bousfield-Kan map). The coend is a weighted limit and this formula for the plain homotopy colimit can be understood the left derived weighted colimit which trivial weight (the underived weight is trivial, but becomes non-trivial after derivation – this extra complexity helps reduce the complexity for the replacement for the functor FF itself).

From the fact that this is a Quillen bifunctor and using the observation that for the trivial weight W=const1W = const 1 the weighted colimit reduces to an ordinary colimit, follows the above Bousfield-Kan-type formula for the homotopy colimit.

Bar constructions

A general way of obtaining resolutions that compute homotopy (co)limits involves bar constructions. (…)

Homotopy limits versus higher categorical limits

As a special case of enriched homotopy theory, we may consider model categories or homotopical categories that are enriched over a notion of nn-category as presentations for (n+1)(n+1)-categories. (Here we allow nn to also be of the form (n,r), with the obvious convention that (n,r)+1=(n+1,r+1)(n,r)+1 = (n+1,r+1) and ∞+1=∞\infty+1=\infty.) For example:

simplicial sets are models for ∞\infty-groupoids ((∞,0)(\infty,0)-categories), so simplicial model categories are presentations for (∞,1)(\infty,1)-categories. Of course, arbitrary model categories are also presentations for (∞,1)(\infty,1)-categories, but simplicial model categories are easier to work with, and in particular easier to construct homotopy limits in.

A (2Cat,⊗Gray(2Cat,\otimes_{Gray})-enriched category is a Gray-category, and a model Gray-category? or homotopical Gray-category? is a presentation of a weak 3-category (or tricategory).

If CC is a category enriched over (n−1)(n-1)-categories and we are considering it to be an nn-category (which happens to be strict at the bottom level), it is natural to define a “weak equivalence” in the underlying ordinary category of CC to be a morphism that is an nn-category-theoretic equivalence. We call this the natural or trivial homotopical structure on CC. In certain cases (such as n=2)n=2) it can be made into a model structure, also called natural or trivial.

Since higher categorical limits are generally defined as representing objects for cones that commute up to equivalence, it is unsurprising that if CC has a natural homotopical structure, locally-defined homotopy limits and nn-limits coincide. For n=1n=1 this is trivial. For n=2n=2 it is proven in (Gambino 10) (particularly section 6). For n=(∞,1)n=(\infty,1) it is proven in (among other places) Lurie’s book, section 4.2.4. The case n=3n=3 ought to be approachable in theory, but doesn’t seem to have been done (probably partly because the general theory of 3-limits is fairly nonexistent).

On the other hand, we may also consider a category CC enriched over nn-categories with a larger class of weak equivalences than just the nn-categorical equivalences. Then CC presents an nn-category (its “homotopy nn-category”) obtained by formally turing these weak equivalences into nn-categorical equivalences. Homotopy limits in CC with this homotopical structure should then present nn-limits in its homotopy nn-category. In the case n=(∞,1)n=(\infty,1) this is also essentially in Lurie’s book; for other values of nn it may not be in the literature.

Homotopy limits versus limits in the homotopy category

It is important to note that homotopy limits and limits in the homotopy category are, in general, incomparable. A homotopy limit need not be a limit in the homotopy category, while a limit in the homotopy category need not be a homotopy limit.

It is generally true that homotopy products (and coproducts) are also products and coproducts in the homotopy category. Some other homotopy limits induce the corresponding notion of weak limit in the homotopy category; for instance, homotopy pullbacks become weak pullbacks in the homotopy category. However, even this is not true for all types of homotopy limit.

On the other hand, homotopy categories do not usually have many limits and colimits at all (aside from products and coproducts). An explicit proof that Ho(Cat)Ho(Cat) does not have pullbacks can be found here. But even if a homotopy category does happen to have limits and colimits, these need not be the same as homotopy limits.

For instance, every chain complex over a fieldkk is quasi-isomorphic to its homology, regarded as a chain complex with zero differentials; and between chain complexes of the latter form, quasi-isomorphisms are just isomorphisms. Thus, the homotopy category of chain complexes over kk is equivalent to the category of graded kk-vector-spaces. This is complete and cocomplete as a category, but its limits and colimits are not the same as the homotopy limits and colimits arising from its presentation as the homotopy category of chain complexes. In particular, chain complexes are a stable (infinity,1)-category, so every homotopy pullback square is also a homotopy pushout square and vice versa; but nothing of the sort is true in graded vector spaces as a 1-category.

Proof

For instance prop 14.8.8 in

Hirschhorn, Model categories and their localization

Notice that if F:Δop→CF : \Delta^{op} \to C takes values in cofibrant objects of CC, then it is itself cofibrant as an object of [Δop,C]inj[\Delta^{op},C]_{inj}. In that case no further cofibrant replacement of FF is necessary and it therefore follows with the general formula and the above proposition that the homotopy colimit over FF is given by the formulas

This is famously the formula introduced and used by Bousfield and Kan (but there originally missing the necessary condition that FF be objectwise cofibrant). See Bousfield-Kan map.

Homotopy pushouts

Let in the above general formula D={a←c→b}D = \{a \leftarrow c \to b\} be the walkingspan. Ordinary colimits parameterized by such DD are pushouts. Homotopy colimits over such DD are homotopy pushouts.

Observation

Observation

For DD as above, a functor F:D→CF : D \to C is cofibrant in [D,C]proj[D,C]_{proj} if

it sends both morphisms c→ac \to a and c→bc \to b to cofibrations

it sends cc (and hence also aa and bb) to cofibrant objects in CC.

Since a coend ∫*⊗F\int {* } \otimes F over a tensor product where the first factor in the integrand in the tensor unit is just an ordinary colimit over the remaining FF, it follows that if FF is of the form of the above observation, then the ordinary colimit over FF already computes the homotopy pushout:

hocolimF=lim→F.
hocolim F = \lim_\to F
\,.

The dual version of this statement (for homotopy limits and homotopy pullbacks) is discussed in more detail in the examples below.

Homotopy pullbacks

Let D={1→0←2}D = \{ 1\to 0 \leftarrow 2\} be the pullback diagram, so that limits over it compute pullbacks, and assume that F:D→CF : D \to C is such that

F(1)→F(0)←F(2)
F(1) \to F(0) \leftarrow F(2)

satisfies * F(i)F(i) is fibrant for all ii; * and either F(1)→F(0)F(1) \to F(0) or F(2)→F(0)F(2) \to F(0) is a fibration;

then

holimDFholim_D F exists;

and is weakly equivalent to the ordinary limit holimDF⟶≃limDFholim_D F \stackrel{\simeq}{\longrightarrow} lim_D F.

Conversely this means that on an arbitrary pullback diagram holimDFholim_D F can be computed by finding a natural transformation F⇒RFF \Rightarrow R F whose component morphisms are weak equivalences and such that RFR F satisfies the above conditions.

Based loop objects

For BB any pointed object with point pt⟶ptBBpt \stackrel{pt_B}{\longrightarrow} B the homotopy pullback of the point along itself is the loop space object of BB

Fibration sequences

the sequence A→B→CA \to B \to C is called a fibration sequence. The object AA is the homotopy kernel or homotopy fiber of B→CB \to C. Since homotopy pullback squares compose to homotopy pullback squares, the homotopy kernel of a homotopy kernel is not trivial, but is a loop space object

Homotopy pullback of a point over a group / universal bundles

As a special case of the above general example we get the following.

Let C=C =Grpd equipped with the canonical model structure. Write GG for a group regarded as a discrete monoidal groupoid (elements of GG are the objects of the groupoids and all morphisms are identities) write and BG\mathbf{B}G for the corresponding one-object groupoid (single object, one morphism per element of GG). Write ptpt for the terminal groupoid (one object, no nontrivial morphism). Notice that there is a unique functor pt→BGpt \to \mathbf{B}G. Then we have

To see this, we compute using the above prescription by finding a weakly equivalent pullback diagram such that one of its morphisms is a fibration. This is achieved in particular by the generalized universal bundlept⟶≃EG→>BGpt \stackrel{\simeq}{\longrightarrow} \mathbf{E}G \to\gt \mathbf{B}G, where EG\mathbf{E}G is the action groupoidG//GG//G of GG acting on itself by multiplication from one side. So we have a weak equivalence of pullback diagrams

and the homotopy limit in question is weakly equivalent to the ordinary limit over the lower diagram. That is directly seen to be Disc(Obj(EG))=Disc(Obj(G//G))=Disc(G)Disc(Obj(\mathbf{E}G)) = Disc(Obj(G//G)) = Disc(G) which we just write as GG:

This example is important in the context of groupoidification and geometric function theory, as described there. A closely related example is the following: a functor ρ:BG→Top\rho:\mathbf{B}G\to {Top} is the datum of a toplogical space XX equipped with an action of GG. Then, colim(ρ)=X/Gcolim(\rho)=X/G whereas hocolim(ρ)=EG×GXhocolim(\rho)=\mathbf{E}G\times_G X, see equivariant cohomology.

Homotopy pullback of a subgroup over a group

The above example generalizes straightforwardly to the case where the trivial inclusion pt→BGpt \to \mathbf{B}G is replaced by any inclusion BH↪BG\mathbf{B}H \hookrightarrow \mathbf{B}G of any subgroup HH of GG pretty much literally by replacing ptpt by BH\mathbf{B}H throughout.

where on the right we have the action groupoid of H×HH \times H acting on GG by multiplication from the left (first factor) and the right (second factor). (See for instance at Hecke category for an application.)

To see this, we again build a fibrant replacement of the pullback diagram. Following the constructions at generalized universal bundle consider first the groupoid EBHG\mathbf{E}_{\mathbf{B}H}G given by the pullback diagram

As at generalized universal bundle one proves that the left vertical morphism EBHG→BG\mathbf{E}_{\mathbf{B}H}G \to \mathbf{B}G is a fibration.

Now, notice (which was implicit in the above example) that since [I,BG][I,\mathbf{B}G] is a path object in a category of fibrant objects we have a sectionBG→≃σ[I,BG]
\mathbf{B}G \stackrel{\simeq}{\to}^\sigma [I, \mathbf{B}G]
of [I,BG]→d0BG[I,\mathbf{B}G] \stackrel{d_0}{\to} \mathbf{B}G. In the above pullback diagram this induces a morphism BH→σEBHG
\mathbf{B}H \stackrel{\sigma}{\to} \mathbf{E}_{\mathbf{B}H}G
making the obvious diagram commute. Now, the latter morphism, being the pullback of an acyclic fibration is an acyclic fibration, so its right inverse σ\sigma is a weak equivalence. This way we obtain the morphism of pullback diagrams

which is objectwise a weak equivalence and such that the horizontal morphism on the bottom left is a fibration. By the above statement the ordinary limit of the lower horizontal diagram is weakly equivalent to the homotopy limit we are looking for. But this is manifestly the desired action groupoid:

where Q′Reedy(⋯)Q'_{Reedy}(\cdots) is a cofibrant resolution in the Reedy model structure [Δ,sSetQuillen]Reedy[\Delta,sSet_{Quillen}]_{Reedy} and QReedy(...)Q_{Reedy}(...) in [Δop,sSetQuillen]Reedy[\Delta^{op}, sSet_{Quillen}]_{Reedy}. But by the discussion at Reedy model structure – simplex category we have that

Remark

More generally with this kind of argument it follows that generally the homotopy colimit over a simplicial diagram of simplicial sets is represented by the diagonal simplicial set of the corresponding bisimplicial set.

Remark

This kind of argument has many immediate generalizations. For instance for C=[Kop,sSetQuillen]injC = [K^{op}, sSet_{Quillen}]_{inj} the injective model structure on simplicial presheaves over any small category KK, or any of its left Bousfield localizations, we have that the cofibrations are objectwise those of simplicial sets, hence objectwise monomorphisms, hence it follows that every simplicial presheaf XX is the hocolim over its simplicial diagram of component presheaves.

For the following write Δ:Δ→sSet\mathbf{\Delta} : \Delta \to sSet for the fat simplex.

Observation

The fat simplex is Reedy cofibrant.

Proof

By the discussion at homotopy colimit, the fat simplex is cofibrant in the projective model structure on functors[Δ,sSetQuillen]proj[\Delta, sSet_{Quillen}]_{proj}. By the general properties of Reedy model structures, the identity functor [Δ,sSetQuillen]proj→[Δ,sSetQuillen]Reedy[\Delta, sSet_{Quillen}]_{proj} \to [\Delta, sSet_{Quillen}]_{Reedy} is a left Quillen functor, hence preserves cofibrant objects.

Proof

(as discussed there). Therefore with its second argument fixed and cofibrant it is a left Quillen functor in the remaining argument. As such, it preserves weak equivalences between cofibrant objects (by the factorization lemma). By the above discussion, both Δ[n]\mathbf{\Delta}[n] and Δ[−]\Delta[-] are indeed cofibrant in [Δ,sSetQuillen]Reedy[\Delta,sSet_{Quillen}]_{Reedy}. Clearly the functor Δ[−]→Δ[−]\mathbf{\Delta}[-] \to \Delta[-] is objectwise a weak equivalence in sSetQuillensSet_{Quillen}, hence is a weak equivalence.

Proposition

Let i:Δf↪Δ i : \Delta_f \hookrightarrow \Delta be the inclusion into the simplex category of all the monomorphisms (all the face maps).

This inclusion is a homotopy-initial functor. As a consequence, homotopy colimits of shape Δ\Delta can equivalently be computed after their restriction to Δf\Delta_f

and using the obvious face and degeneracy maps: face maps act by mapping components of the coproducts of one sequence of morphisms to one obtained by deleting outer arrows or composing inner arrows. If the rightmost arrow is deleted, then the component map is not the identity but is F(dn)→D(dn−1)F(d_n) \to D(d_{n-1}). The degeneracy maps similarly introduce identity morphisms.

of simplicial (pre)sheaves. One would like this to extend to a Quillen adjunction that recalls the fact that it came from a geometric morphism by the fact that the left adjointinverse image functor SSh(C′)→SSh(C)SSh(C') \to SSh(C) preserves finite homotopy limits.

is given a global definition of homotopy (co)limit as 4.1, p. 14, and it is discussed how to compute homotopy (co)limits concretely using local constructions. For instance the above statement on the computation of homotopy pullbacks is proposition 2.5, p. 15

A nice discussion of the expression of homotopy colimits in terms of coends is in